High-force density three pole magnetic bearing

ABSTRACT

A first radial force value and a second radial force value is received by a radial magnetic bearing controller. Coefficients are computed for a first equation using the first and second radial force values. The first equation is solved to define first solution values. A second solution value paired with each first solution value is computed using the first radial force value and a respective first solution value to define second solution values. Control current sets are computed for each unique paired solution of the second solution values and the first solution values. Each control current set includes a control current value for each of three control currents. A control current value for each of the three control currents is selected from the control current sets. The control current value for each of the three control currents is output to a respective radial winding of a three-pole radial magnetic bearing.

BACKGROUND

Mechanical bearings used to support the shaft of high speed motorsystems severely limit the system lifetime, can be a source ofsignificant losses, and require lubricants that can interfere with thebroader system. Active magnetic bearings (AMBs) offer a contact-freesolution to overcome these issues. This technology has seen commercialdeployment in certain compressor applications, such as HVAC chillers,wastewater aeration, and natural gas transportation. Recently,bearingless motors have been developed that combine the electric motorand the AMB into a single machine where the same iron and copper areused for creating torque and suspension forces resulting in a highlyintegrated, compact, and low cost design with the potential to overcomethe shortcomings of AMB-based motor systems.

A traditional bearingless motor produces radial suspension forces withinthe motor's air gap. This means that it is capable of stabilizing tworadial degrees of freedom (DOF), but relies on external support for theremaining three DOF (axial and tilting directions). This can be providedby using standard radial AMBs and axial AMBs, but the bulky nature ofthese devices limit the shaft length available to the motor becauseportions of the shaft length are occupied by the motor bearings and aretherefore unavailable to be used for creating torque.

SUMMARY

In an example embodiment, a non-transitory computer-readable medium isprovided having stored thereon computer-readable instructions that, whenexecuted by a processor, cause a controller to compute a current foreach radial coil of a radial magnetic bearing. An indicator of a firstradial force value in a first radial force direction and a second radialforce value in a second radial force direction is received. The firstradial force direction is perpendicular to the second radial forcedirection. Coefficients are computed for a first equation using thefirst radial force value and the second radial force value. The firstequation is a non-linear equation. The first equation is solved todefine a plurality of first solution values. A second solution valuepaired with each first solution value of the plurality of first solutionvalues is computed using the first radial force value and a respectivefirst solution value to define a plurality of second solution values. Aplurality of control current sets is computed. Each control current setof the plurality of control current sets includes a control currentvalue for each of three control currents. The plurality of controlcurrent sets includes a control current set computed for each uniquepaired solution of the plurality of second solution values and theplurality of first solution values. A control current value for each ofthe three control currents is selected from a control current set of thecomputed plurality of control current sets. The selected control currentvalue for each of the three control currents is output to a respectiveradial winding of a three-pole radial magnetic bearing.

Other principal features of the disclosed subject matter will becomeapparent to those skilled in the art upon review of the drawingsdescribed below, the detailed description, and the appended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

Illustrative embodiments of the disclosed subject matter will hereafterbe described referring to the accompanying drawings, wherein likenumerals denote like elements.

FIG. 1 depicts a motor system in accordance with an illustrativeembodiment.

FIG. 2 depicts a perspective view of a combined radial-axial magneticbearing (CRAMB) of the motor system of FIG. 1 in accordance with anillustrative embodiment.

FIG. 3 depicts an exploded, perspective view of the CRAMB of FIG. 2 inaccordance with an illustrative embodiment.

FIG. 4 depicts a perspective view of the CRAMB of FIG. 2 withoutexternal components to view an interior arrangement of the CRAMB inaccordance with an illustrative embodiment.

FIG. 5 depicts a perspective view of the CRAMB of FIG. 2 withoutadditional components in accordance with an illustrative embodiment.

FIG. 6 depicts a perspective view of the CRAMB of FIG. 2 without stilladditional components in accordance with an illustrative embodiment.

FIG. 7 depicts a perspective view of the CRAMB of FIG. 2 without yetadditional components in accordance with an illustrative embodiment.

FIG. 8 depicts a perspective view of a radial rotor lamination stack, aradial stator lamination stack, and radial coils of the CRAMB of FIG. 2in accordance with an illustrative embodiment.

FIG. 9 depicts a top view of the radial rotor lamination stack, theradial stator lamination stack, and the radial coils of FIG. 8 inaccordance with an illustrative embodiment.

FIG. 10 depicts a simplified top view of the radial rotor laminationstack, the radial stator lamination stack, and the radial coils of FIG.8 with an illustrative orientation and with illustrative dimensions.

FIG. 11 depicts a center axial plane cross-sectional view of the CRAMBof FIG. 2 in accordance with an illustrative embodiment.

FIG. 12 depicts a top radial plane view of the CRAMB of FIG. 2 inaccordance with an illustrative embodiment.

FIG. 13 depicts a bottom radial plane view of the CRAMB of FIG. 2 inaccordance with an illustrative embodiment.

FIG. 14 depicts a bottom radial plane view of the CRAMB of FIG. 2 with abaseplate removed in accordance with an illustrative embodiment.

FIG. 15 depicts a bottom radial plane view of the CRAMB of FIG. 2without the external components in accordance with an illustrativeembodiment.

FIG. 16 depicts an axial view of a rotor of the CRAMB of FIG. 2 inaccordance with an illustrative embodiment.

FIG. 17 depicts a perspective view of an axial stator cap of the CRAMBof FIG. 2 in accordance with an illustrative embodiment.

FIG. 18 depicts a perspective view of an axial stator of the CRAMB ofFIG. 2 in accordance with an illustrative embodiment.

FIG. 19 depicts a perspective view of an axial coil of the CRAMB of FIG.2 in accordance with an illustrative embodiment.

FIG. 20 depicts a perspective view of a permanent magnet of the CRAMB ofFIG. 2 in accordance with an illustrative embodiment.

FIG. 21 depicts a perspective view of a compensation coil of the CRAMBof FIG. 2 in accordance with an illustrative embodiment.

FIG. 22 depicts a first perspective view of a first radial coil of theCRAMB of FIG. 2 in accordance with an illustrative embodiment.

FIG. 23 depicts a second perspective view of the first radial coil ofFIG. 22 in accordance with an illustrative embodiment.

FIG. 24 depicts a perspective view of a ring stator of the CRAMB of FIG.2 in accordance with an illustrative embodiment.

FIG. 25 depicts a top view of the ring stator of FIG. 24 in accordancewith an illustrative embodiment.

FIG. 26 depicts a bottom view of the radial stator lamination stack ofthe CRAMB of FIG. 2 in accordance with an illustrative embodiment.

FIG. 27 depicts a perspective view of the radial rotor lamination stackof the CRAMB of FIG. 2 in accordance with an illustrative embodiment.

FIG. 28 depicts a perspective view of a standoff of the CRAMB of FIG. 2in accordance with an illustrative embodiment.

FIG. 29 depicts a top perspective view of a baseplate of the CRAMB ofFIG. 2 in accordance with an illustrative embodiment.

FIG. 30 depicts a flow diagram illustrating examples of operationsperformed by a radial magnetic bearing controller of a radial magneticbearing portion of the CRAMB of FIG. 2 in accordance with anillustrative embodiment.

FIG. 31 depicts a normalized rated force as a function of a normalizedbias field computed for the radial magnetic bearing portion of the CRAMBof FIG. 2 in accordance with an illustrative embodiment.

FIG. 32 depicts a maximum rated force vector of a three pole bearing andshows the individual force components that make up the net force vectorin accordance with an illustrative embodiment.

DETAILED DESCRIPTION

Referring to FIG. 1, a block diagram of an electrical machine system 100is shown in accordance with an illustrative embodiment. Electricalmachine system 100 may include a motor 102, a combined radial and axialmagnetic bearing (CRAMB) 104, a radial magnetic bearing (RMB) 106, arotor 108, and an electrical machine controller 110. Rotor 108 is commonto motor 102, CRAMB 104, and RMB 106.

Motor 102 can be implemented as any type of radial flux machine such asa permanent magnet (PM) machine, a synchronous reluctance machine, aninduction machine, a consequent-pole machine, an alternating current(AC) homopolar machine, etc. Motor 102 further may be bearingless motoreliminating a need for RMB 106. A bearingless motor provides afunctionality of a magnetic bearing and a motor in a single electricmachine. Compared to systems that utilize a motor with separate magneticbearings, bearingless technology results in a more integrated systemthat requires less raw material and can be designed for higher speedsdue to shorter shaft lengths. Applications for a bearingless motor rangefrom low speed, hygienic mixing devices, pumps, and artificial hearts tohigh and ultra-high speed machines for flywheels, spindle tools, andturbomachinery, etc.

Electrical machine controller 110 may include an input interface 112, anoutput interface 114, a non-transitory computer-readable medium 116, aprocessor 118, and a control application 120. Fewer, different, and/oradditional components may be incorporated into electrical machinecontroller 110.

Input interface 112 provides an interface for receiving information fromthe user or another device for entry into electrical machine controller110 as understood by those skilled in the art. Input interface 112 mayinterface with various input technologies including, but not limited to,a keyboard, a microphone, a mouse, a display, a track ball, a keypad,one or more buttons, etc. to allow the user to enter information intoelectrical machine controller 110 or to make selections presented in auser interface displayed on the display. Input interface 112 may furtherreceive signals such as sensor signals from any of motor 102, CRAMB 104,and RMB 106.

The same interface may support both input interface 112 and outputinterface 114. For example, a touch screen provides a mechanism for userinput and for presentation of output to the user. Electrical machinecontroller 110 may have one or more input interfaces that use the sameor a different input interface technology. The input interfacetechnology further may be accessible by electrical machine controller110 through a communication interface (not shown).

Output interface 114 provides an interface for outputting informationfor review by a user of electrical machine controller 110 and/or for useby another application or device. For example, output interface 114 mayinterface with various output technologies including, but not limitedto, the display, a speaker, a printer, etc. Electrical machinecontroller 110 may have one or more output interfaces that use the sameor a different output interface technology. Output interface 114 mayfurther output control signals in the form of currents or voltages toany of motor 102, CRAMB 104, and RMB 106. Output interface 114 furthermay be accessible by electrical machine controller 110 through thecommunication interface.

Computer-readable medium 116 is an electronic holding place or storagefor information so the information can be accessed by processor 118 asunderstood by those skilled in the art. Computer-readable medium 116 caninclude, but is not limited to, any type of random access memory (RAM),any type of read only memory (ROM), any type of flash memory, etc. suchas magnetic storage devices (e.g., hard disk, floppy disk, magneticstrips, . . . ), optical disks (e.g., compact disc (CD), digitalversatile disc (DVD), . . . ), smart cards, flash memory devices, etc.Electrical machine controller 110 may have one or more computer-readablemedia that use the same or a different memory media technology. Forexample, computer-readable medium 116 may include different types ofcomputer-readable media that may be organized hierarchically to provideefficient access to the data stored therein as understood by a person ofskill in the art. As an example, a cache may be implemented in asmaller, faster memory that stores copies of data from the mostfrequently/recently accessed main memory locations to reduce an accesslatency. Electrical machine controller 110 also may have one or moredrives that support the loading of a memory media such as a CD, DVD, anexternal hard drive, etc. One or more computer-readable media may beconnected to electrical machine controller 110 using the communicationinterface.

Processor 118 executes instructions as understood by those skilled inthe art. The instructions may be carried out by a special purposecomputer, logic circuits, or hardware circuits. Processor 118 may beimplemented in hardware and/or firmware. Processor 118 executes aninstruction, meaning it performs/controls the operations called for bythat instruction. The term “execution” is the process of running anapplication or the carrying out of the operation called for by aninstruction. The instructions may be written using one or moreprogramming language, scripting language, assembly language, etc.Processor 118 operably couples with input interface 112, with outputinterface 114, and with computer-readable medium 116 to receive, tosend, and to process information. Processor 118 may retrieve a set ofinstructions from a permanent memory device and copy the instructions inan executable form to a temporary memory device that is generally someform of RAM. Electrical machine controller 110 may include a pluralityof processors that use the same or a different processing technology.

Some processors may be central processing units (CPUs). Some processesmay be more efficiently and speedily executed and processed with machinespecific processors. For example, some of these processors can includean application-specific integrated circuit, a field-programmable gatearray, a purpose-built chip architecture, etc. using semiconductordevices.

Control application 120 performs operations associated with controllingoperation of motor 102, CRAMB 104, and/or RMB 106. Some or all of theoperations described herein may be embodied in control application 120.The operations may be implemented using hardware, firmware, software, orany combination of these methods. Referring to the example embodiment ofFIG. 1, control application 120 is implemented in software (comprised ofcomputer-readable and/or computer-executable instructions) stored incomputer-readable medium 116 and accessible by processor 118 forexecution of the instructions that embody the operations of controlapplication 120. Control application 120 may be written using one ormore programming languages, assembly languages, scripting languages,etc.

Referring to FIG. 2, a perspective view of CRAMB 104 is shown inaccordance with an illustrative embodiment. Referring to FIG. 3, anexploded, perspective view of CRAMB 104 is shown in accordance with anillustrative embodiment. CRAMB 104 may include an axial stator cap 200,an axial stator 202, a plurality of permanent magnets 204, a ring stator206, a baseplate 208, a plurality of standoffs 210, a radial statorlamination stack 300, a compensation coil 302, three radial coils 304,an axial coil 400 (shown referring to FIG. 4), and a radial rotorlamination stack 700 (shown referring to FIG. 7).

In the illustrative embodiment, CRAMB 104 has a side-by-side topologythat includes a three pole RMB portion (stage) that is separated from anaxial magnetic bearing (AMB) portion (stage) by the plurality ofpermanent magnets 204. The plurality of permanent magnets 204 provide abias flux that magnetizes CRAMB 104, while the RMB portion and the AMBportion each contain the three radial coils 304 and axial coil 400,respectively, to create controllable radial and axial forces on rotor108, respectively. Though not shown, RMB 106 can be configured similarto the three pole RMB portion of CRAMB 104 to exert radial forces on anopposite end of rotor 108. The side-by-side topology of CRAMB 104reduces a required shaft length of rotor 108 by increasing both a radiallinear force density and an axial linear force density.

CRAMB 104 includes compensation coil 302 housed in the RMB portion ofCRAMB 104 to decouple the axial and radial operation. Compensation coil302 is connected in series with axial coil 400 to compensate a parasiticflux produced by axial coil 400.

The components of CRAMB 104 and RMB 106 may be formed of one or morematerials, such as a metal, a magnetic material, and/or a plastic havinga sufficient strength and rigidity and conductivity to provide thedescribed function. In an illustrative embodiment, components of CRAMB104 and RMB 106 are formed of a metal material such as steel.

Referring to FIG. 4, a perspective view of CRAMB 104 without axialstator cap 200, axial stator 202, ring stator 206, baseplate 208, andthe plurality of standoffs 210 is shown in accordance with anillustrative embodiment. Referring to FIG. 5, a perspective view ofCRAMB 104 without axial stator cap 200, axial stator 202, ring stator206, baseplate 208, the plurality of standoffs 210, and axial coil 400is shown in accordance with an illustrative embodiment. Referring toFIG. 6, a perspective view of CRAMB 104 without axial stator cap 200,axial stator 202, ring stator 206, baseplate 208, the plurality ofstandoffs 210, axial coil 400, and radial stator lamination stack 300 isshown in accordance with an illustrative embodiment. Referring to FIG.7, a perspective view of CRAMB 104 without axial stator cap 200, axialstator 202, ring stator 206, baseplate 208, the plurality of standoffs210, axial coil 400, radial stator lamination stack 300, compensationcoil 302, and the three radial coils 304 is shown in accordance with anillustrative embodiment.

Referring to FIG. 8, a perspective view of radial rotor lamination stack700, radial stator lamination stack 300, and the three radial coils 304are shown in accordance with an illustrative embodiment. Referring toFIG. 9, a top view of radial rotor lamination stack 700, radial statorlamination stack 300, and the three radial coils 304 are shown inaccordance with an illustrative embodiment. Referring to FIG. 10, asimplified top view of radial rotor lamination stack 700, radial statorlamination stack 300, and the three radial coils 304 are shown with anillustrative orientation and with illustrative dimensions. Radial statorlamination stack 300 includes a first radial stator lamination tooth802, a second radial stator lamination tooth 804, and a third radialstator lamination tooth 806. The three radial coils 304 include a firstradial coil 808, a second radial coil 810, and a third radial coil 812.First radial coil 808, second radial coil 810, and third radial coil 812define a shape having N turns of coil winding wound around first radialstator lamination tooth 802, second radial stator lamination tooth 804,and third radial stator lamination tooth 806, respectively. Radialstator lamination stack 300 has a stator lamination stack thickness 814in the axial direction. Radial rotor lamination stack 700 and radialstator lamination stack 300 are radially separated by a radial air gaphaving a radial air gap length 900. First radial stator lamination tooth802, second radial stator lamination tooth 804, and third radial statorlamination tooth 806 are shaped to have a radial tooth angle 1000 thatdefines an angle that each radial tooth spans relative to a rotor shaftcenter 1002. Radial stator lamination stack 300 may be shaped such thata stator lamination stack aperture wall 2600 (shown referring to FIG.26) has a stator inner radius 1004 measured from rotor shaft center1002. An x-axis and a y-axis define a radial plane of radial statorlamination stack 300.

Referring to FIG. 11, a center axial plane cross-sectional view of CRAMB104 is shown in accordance with an illustrative embodiment. Referring toFIG. 12, a top radial plane view of CRAMB 104 is shown in accordancewith an illustrative embodiment. Referring to FIG. 13, a bottom radialplane view of CRAMB 104 is shown in accordance with an illustrativeembodiment. Referring to FIG. 14, a bottom radial plane view of CRAMB104 is shown with baseplate 108 removed in accordance with anillustrative embodiment. Referring to FIG. 15, a bottom radial planeview of CRAMB 104 is shown without the external components in accordancewith an illustrative embodiment.

Referring to FIG. 16, an axial view of rotor 108 is shown in accordancewith an illustrative embodiment. Rotor 108 may be formed of one or moreconductive materials such as steel. Rotor 108 is positioned radiallyinterior of CRAMB 104, motor 102, and/or RMB 106 and has a generallycircular cross section with varying radius to provide an air gap betweenthe various components of CRAMB 104, motor 102, and/or RMB 106. Forexample, rotor 108 may include a rotor shaft 1600, a rotor cap portion1602, a rotor axial portion 1604, a rotor compensation portion 1606, anda rotor radial portion 1608.

The RMB portion of CRAMB 104 may include radial stator lamination stack300, the three radial coils 304, radial rotor lamination stack 700, androtor radial portion 1608. The AMB portion of CRAMB 104 may includeaxial stator 202 and rotor axial portion 1604. Compensation coil 302 andthe plurality of permanent magnets 204 function as parts of both the RMBportion and the AMB portion.

Referring to FIG. 17, a perspective view of axial stator cap 200 isshown in accordance with an illustrative embodiment. Axial stator cap200 may include a stator cap shaft inner wall 1700 and a plurality ofcap standoff aperture walls 1702. Rotor cap portion 1602 is sized andshaped to fit within stator cap shaft inner wall 1700.

Referring to FIG. 18, a perspective view of axial stator 202 is shown inaccordance with an illustrative embodiment. Axial stator 202 may includea stator base 1800, a stator shaft inner wall 1802, and a stator outerwall 1804. Stator inner aperture wall 1802 is formed through stator base1800. Stator inner aperture wall 1802 extends from stator base 1800toward axial stator cap 200 when axial stator 202 and axial stator 202are mounted to CRAMB 104. Rotor axial portion 1604 is sized and shapedto fit within stator inner aperture wall 1802.

Referring to FIG. 19, a perspective view of axial coil 400 is shown inaccordance with an illustrative embodiment. Axial coil 400 may includean axial coil inner wall 1900 and an axial coil outer wall 1902. Rotoraxial portion 1604 fits within axial coil inner wall 1900.

Referring to FIG. 20, a perspective view of a permanent magnet 2000 ofthe plurality of permanent magnets 204 is shown in accordance with anillustrative embodiment. Permanent magnet 2000 may be formed of amagnetic material selected to have a PM width 2002, a PM depth 2004, aPM height 2006 that may be the same or different to provide a selectedbias field B₀. In an illustrative embodiment, the plurality of permanentmagnets 204 include 48 permanent magnet cubes. IN an alternativeembodiment, the plurality of permanent magnets 204 may be replaced witha continuous ring magnet.

Referring to FIG. 21, a perspective view of compensation coil 302 isshown in accordance with an illustrative embodiment. Compensation coil302 may include a compensation coil inner wall 2100 sized and shapedsuch that rotor compensation portion 1606 fits therein and compensationcoil outer wall 2100 sized and shaped to within first radial coil 808,second radial coil 810, and third radial coil 812 above radial statorlamination stack 300. Compensation coil 302 may be formed of a coilconsisting of many turns of an electrically conductive material such ascopper, aluminum, etc.

Referring to FIG. 22, a first perspective view of first radial coil 808is shown in accordance with an illustrative embodiment. Referring toFIG. 23, a second perspective view of first radial coil 808 is shown inaccordance with an illustrative embodiment. Second radial coil 810 andthird radial coil 812 are wound to form a shape similar to first radialcoil 808. Though shown as having a solid shape, the shapes illustratedshow a winding shape formed when a first winding is wrapped around firstradial stator lamination tooth 802, when a second winding is wrappedaround second radial stator lamination tooth 804, and when a thirdwinding is wrapped around third radial stator lamination tooth 806. Thewinding shape of first radial coil 808 can be described as defining afirst coil plane 2200, a second coil plane 2202, a third coil plane 2204parallel to first coil plane 2200, and a fourth coil plane 2206 parallelto second coil plane 2202.

The first winding, the second winding, and the third winding are heldwithin the three slots, respectively, and carry a current between aplurality of connectors (not shown in FIGS. 1-3) also called terminals.The windings are wound around the three teeth using various techniquesto form three poles. The first winding, the second winding, and thethird winding carry electrical current as determined below to provide aforce to maintain rotor 108 radially within the radial portion of CRAMB104 and within RMB 106.

Referring to FIG. 24, a perspective view of ring stator 206 is shown inaccordance with an illustrative embodiment. Referring to FIG. 25, a topview of ring stator 206 is shown in accordance with an illustrativeembodiment. Ring stator 206 may include an inner support wall 2400, astator support ledge 2402, an outer support wall 2404, and a pluralityof ring stator standoff aperture walls 2406. Inner support wall 2400 hasa smaller circumference than outer support wall 2404 and extends inwardfrom outer support wall 2404. Radial stator lamination stack 300 ismounted on stator support ledge 2402 that extends above inner supportwall 2400 towards outer support wall 2404 to form a ledge. Axial coilouter wall 1902 is sized and shaped to fit within an inner surface ofouter support wall 2404 and to rest on stator support ledge 2402. Theplurality of ring stator standoff aperture walls 2406 are formed throughprotrusions from outer support wall 2404. Ring stator 206 may be formedof a solid piece of ferromagnetic material such as iron, cobalt, nickel,etc.

Referring to FIG. 26, a bottom view of radial stator lamination stack300 is shown in accordance with an illustrative embodiment. Radialstator lamination stack 300 may include stator lamination stack aperturewall 2600 sized and shaped such that radial rotor lamination stack 800fits therein separated by radial air gap 900. Radial stator laminationstack 300 further may include a first radial coil aperture wall 2602, asecond radial coil aperture wall 2604, a third radial stator coilaperture wall 2606, a first stator neck aperture wall 2608, a secondstator neck aperture wall 2610, a third stator neck aperture wall 2612,a fourth stator neck aperture wall 2614, a fifth stator neck aperturewall 2616, and a sixth stator neck aperture wall 2618.

Stator lamination stack aperture wall 2600, first radial coil aperturewall 2602, second radial coil aperture wall 2604 third radial statorcoil aperture wall 2606, first stator neck aperture wall 2608, secondstator neck aperture wall 2610, third stator neck aperture wall 2612,fourth stator neck aperture wall 2614, fifth stator neck aperture wall2616, and sixth stator neck aperture wall 2618 form a continuousaperture wall formed through each lamination of radial stator laminationstack 300 to form three teeth, first radial stator lamination tooth 802,second radial stator lamination tooth 804, and third radial statorlamination tooth 806, and three slots between each pair of the threeteeth to define the three-poles of RMB 106 and the radial portion ofCRAMB 104. A shape of each slot of the three slots may vary from thatshown in the illustrative embodiment.

As understood by a person of skill in the art, radial stator laminationstack 300 may be formed of laminations mounted closely together andstacked in a z-direction that is perpendicular to the x-axis and they-axis of FIG. 10 to form a right-handed coordinate reference frame andthat extends axially through RMB 106 and CRAMB 104. Each lamination maybe cut to define the three teeth and radial air gap 900 between an outerrotor lamination stack aperture wall 2702 (shown referring to FIG. 27)of radial rotor lamination stack 700 and stator lamination stackaperture wall 2600. Each lamination of stator lamination stack aperturewall 2600 may be formed of a ferromagnetic material such as iron,cobalt, nickel, etc.

Compensation coil 302 is mounted above first stator neck aperture wall2608, second stator neck aperture wall 2610, third stator neck aperturewall 2612, fourth stator neck aperture wall 2614, fifth stator neckaperture wall 2616, and sixth stator neck aperture wall 2618 adjacentfirst radial coil 808, second radial coil 810, and third radial coil812.

Second coil plane 2202 of first radial coil 808 and fourth coil plane2206 of second radial coil 810 wrap around opposite edges of firstradial coil aperture wall 2602. First coil plane 2200 of first radialcoil 808 wraps across a bottom surface of first radial stator laminationtooth 802, and third coil plane 2204 of first radial coil 808 wrapsacross a top surface of first radial stator lamination tooth 802. Secondcoil plane 2202 of second radial coil 810 and fourth coil plane 2206 ofthird radial coil 812 wrap around opposite edges of second radial coilaperture wall 2604. First coil plane 2200 of second radial coil 810wraps across a bottom surface of second radial stator lamination tooth804, and third coil plane 2204 of second radial coil 810 wraps across atop surface of second radial stator lamination tooth 804. Second coilplane 2202 of third radial coil 812 and fourth coil plane 2206 of firstradial coil 808 wrap around opposite edges of third radial stator coilaperture wall 2606. First coil plane 2200 of third radial coil 812 wrapsacross a bottom surface of third radial stator lamination tooth 806, andthird coil plane 2204 of third radial coil 812 wraps across a topsurface of third radial stator lamination tooth 806.

Referring to FIG. 27, a perspective view of radial rotor laminationstack 700 is shown in accordance with an illustrative embodiment. Radialrotor lamination stack 700 may include an inner rotor lamination stackaperture wall 2700 and outer rotor lamination stack aperture wall 2702.Inner rotor lamination stack aperture wall 2700 is sized and shaped suchthat rotor radial portion 1608 fits therein. Outer rotor laminationstack aperture wall 2702 fits within stator lamination stack aperturewall 2600 separated by radial air gap 900. As understood by a person ofskill in the art, radial rotor lamination stack 700 may be formed oflaminations mounted closely together and stacked in the z-direction.Each lamination may be cut to define inner rotor lamination stackaperture wall 2700 and outer rotor lamination stack aperture wall 2702.Each lamination of radial rotor lamination stack 700 may be formed of aferromagnetic material such as iron, cobalt, nickel, etc.

Referring to FIG. 28, a perspective view of a standoff 2800 of theplurality of standoffs 210 is shown in accordance with an illustrativeembodiment. Standoff 2800 may include a fastener aperture wall 2802within which a fastener such as a screw or a rivet is inserted to mountthe components of CRAMB 104 together. For example, the screw or therivet is inserted through the plurality of cap standoff aperture walls1702, fastener aperture wall 2802, and the plurality of ring statorstandoff aperture walls 2406 to mount ring stator 206 to axial statorcap 200.

Referring to FIG. 29, a top perspective view of baseplate 208 is shownin accordance with an illustrative embodiment. In an illustrativeembodiment, baseplate 208 is formed of aluminum. Baseplate 208 mayinclude a baseplate base 2900, baseplate cooling aperture walls 2902, abaseplate center plateau 2904, and a rotor aperture wall 2906. Baseplatecooling aperture walls 2902 are formed through baseplate base 2900 toallow heat to escape from CRAMB 104. Baseplate center plateau 2904extends outward from baseplate base 2900. When CRAMB 104 is assembled,baseplate center plateau 2904 is mounted towards an interior of CRAMB104. Rotor radial portion 1608 is formed in baseplate center plateau2904 and is sized and shaped such that rotor aperture wall 2906 fitstherein.

The plurality of permanent magnets 204 create a magnetomotive forceresulting in flux that flows axially through stator side wall 1804 andradially across axial stator cap 200 and stator base 1800 around axialcoil 400, radially through rotor axial portion 1604, axially throughrotor compensation portion 1606 and rotor radial portion 1608, radiallythrough radial rotor lamination stack 700 and radial stator laminationstack 300, and axially through outer support wall 2404 to create biasfield B₀. The flux further flows axially through first coil plane 2200,radial stator lamination stack 300, and third coil plane 2204 of each offirst radial stator lamination tooth 802, second radial statorlamination tooth 804, and third radial stator lamination tooth 806.

Equation (1) below relates a force output of RMB 106 or the radialportion of CRAMB 104 to input control currents:

=k ₁ k ₂·(k ₂

+2B ₀·

)  (1)

where k₁ is a geometry dependent first proportionality constant, k₂ is asecond proportionality constant relating a current to the control fieldit produces, and B₀ is a bias field. The two complex current spacevectors,

and

are defined by equations (2) and (3) for the three-pole RMB portion ofCRAMB 104 and/or of RMB 106 implemented using three poles:

$\begin{matrix}{\overset{harpoonup}{\iota_{c}} = {i_{c1} + {i_{c2} \cdot e^{{- j}\frac{2}{3}\pi}} + {i_{c3} \cdot e^{{- j}\frac{4}{3}\pi}}}} & (2) \\{\overset{harpoonup}{\iota_{c}^{2}} = {i_{c1}^{2} + {i_{c2}^{2} \cdot e^{{- j}\frac{2}{3}\pi}} + {i_{c3}^{2} \cdot e^{{- j}\frac{4}{3}\pi}}}} & (3)\end{matrix}$

In equations 2 and 3, i_(c1), i_(c2), and i_(c3) are calculated bysubtracting off an average of all of the pole currents from the currentsof each pole, for example, using i_(c1)=i₁−i₀, i_(c2)=i₂−i₀,i_(c3)=i₃−i₀, where

${i_{0} = \frac{i_{1} + i_{2} + i_{3}}{3}}.$

Calculation of a force vector given the currents, i₁, i₂, and i₃provided to the first winding wound about first radial stator laminationtooth 802, the second winding wound about second radial statorlamination tooth 804, and the third winding wound about third radialstator lamination tooth 806, respectively, can be computed by computingthe two complex current space vectors,

and

and substituting the computed

and

into equation (1). This formulation applies to magnetic bearings wherethe currents, i₁, i₂, and i₃ sum to zero amperes (A). For designs wherethe currents, i₁, i₂, and i₃ do not sum to zero, the relation applies,but a common-mode (or zero-sequence) current component contributes tothe bias field B₀. To control a magnetic bearing, equation (1) isinverted to compute the currents for the three-pole RMB from the desiredbearing forces using an exact solution to equation (1). Any desiredforce vector can be split into x and y components (F_(x) and F_(y)) thatare used as input with the x-axis and the y-axis defining the radialplane as illustrated in FIG. 10. Assuming ideal field behavior (nofringing fluxes, even field distribution within radial air gap 900 thatcontains only a radial component), the first proportionality constant k₁can be computed as:

$\begin{matrix}{k_{1} = \frac{\beta A}{\mu_{0}}} & (4) \\{\beta = \frac{2{\sin ( \frac{\theta_{1}}{2} )}}{\theta_{1}}} & (5)\end{matrix}$

where β is an area correction factor that accounts for a curvature ofeach tooth, θ₁ is effective radial tooth angle 1000 in radians, A is aneffective air gap area, and μ₀ is a permeability of free space. Forillustration, the air gap area can be computed as A=θ₁RT, where R isstator inner radius 1004, and T is stator lamination stack thickness 814when ideal field behavior is assumed. The second proportionalityconstant k₂ relating currents to control fields can be computed as:

$\begin{matrix}{k_{2} = \frac{\mu_{0}N}{g}} & (6)\end{matrix}$

where g is an effective radial air gap length 900 and N is a number ofturns of first radial coil 808, second radial coil 810, and third radialcoil 812 that may be provided as input values for the case where therotor is centered within the radial air gap.

To generalize this solution for control models where the air gap fieldsand the coil currents have a more complicated relationship (i.e., k₂ isa function of rotor displacement), a modified version of Equation 1 isinverted that yields a solution for air gap fields. Later, the resultingfields are converted to coil currents. The modified version of Equation1 is a function of the air gap fields in front of each radial statorlamination tooth:

=k₁·(B₁ ²+B₂ ²e^(j2/3π)+B₃ ²e^(j4/3π))=k₁·(

+2B₀·

), where

=B_(c1)+B_(c2)·e^(j2/3π)+B_(c3)·e^(j4/3π) and

=B_(c1) ²+B_(c2) ²·e^(j2/3π)e¹⁷+B_(c1) ²+B_(c2) ²·e^(j2/3π)+B_(c3)²·e^(j4/3π). When B₁ is the air gap field in front of first radialstator lamination tooth 802, B₂ is the air gap field in front of secondradial stator lamination tooth 804, and B₃ is the air gap field in frontof third radial stator lamination tooth 806. When B₁ is the air gapfield in front of first radial stator lamination tooth 802, B₂ is theair gap field in front of second radial stator lamination tooth 804, andB₃ is the air gap field in front of third radial stator lamination tooth806, B_(c1), B_(c2), and B_(c3) are calculated by subtracting off theaverage of all of the air gap fields from the air gap in front of eachpole, for example, using B_(c1)=B₁−B₀, B_(c2)=B₂−B₀, B_(c3)=B₃−B₀where

$B_{0} = {\frac{B_{1} + B_{2} + B_{3}}{3}.}$

A control field space vector can be defined as a complex number b_(c)=

with real component x and imaginary component y:

b _(c) =x+yj  (7)

The y-component of equation (7) can be determined by computing the rootsof a depressed quartic polynomial in equation (8), where the forcecomponents F_(x) and F_(y) are included in the quartic coefficients.

y ⁴ +py ² +qy+r=0  (8)

Several well-known methods can be used to compute the roots of adepressed quartic equation. For example, Ferrari's method as describedin Ron Irving Ron, Beyond the Quadratic Formula, American MathematicalSociety, 2010 available through ProQuest Ebook Central can be used.Another common method is to compute eigenvalues of a companion matrix ofthe associated quartic. The quartic coefficients can be computed usingequations (9), (10), and (11).

$\begin{matrix}{p = {3( {\frac{F_{x}}{k_{1}} - {9B_{0}^{2}}} )}} & (9) \\{q = {18B_{0}\frac{F_{y}}{k_{1}}}} & (10) \\{r = {{- \frac{9}{4}}( \frac{F_{y}}{k_{1}} )^{2}}} & (11)\end{matrix}$

Because there are four roots, there are four possible y-component valuesas solutions. The x-component value for each of the y-component valuescan be computed using equation (12).

$\begin{matrix}{x = {\frac{3}{2}( {{2B_{0}} - {\frac{F_{y}}{k_{1}}( \frac{1}{y} )}} )}} & (12)\end{matrix}$

When a y-component value of the four possible y-component values iszero, a depressed quartic polynomial in terms of x can be definedinstead using equation (13).

x ⁴ px ² qx+r=0  (13)

where the quartic coefficients can be computed using equations (14),(15), and (16).

$\begin{matrix}{p = {{- 3}( {\frac{F_{x}}{k_{1}} + {9B_{0}^{2}}} )}} & (14) \\{q = {18{B_{0}( {{3B_{0}^{2}} + \frac{F_{x}}{k_{1}}} )}}} & (15) \\{r = {- ( {{27{B_{0}^{2}( \frac{F_{x}}{k_{1}} )}} + {\frac{9}{4}( \frac{F_{y}}{k_{1}} )^{2}}} )}} & (16)\end{matrix}$

Again, because there are four roots, there are four possible x-componentvalues as solutions. The y-component value for each of the x-componentvalues can be computed using equation (17).

$\begin{matrix}{y = {\frac{F_{y}}{k_{1}}( {{2B_{0}} - {\frac{2}{3}x}} )^{- 1}}} & (17)\end{matrix}$

Either method results in the same values for x and y, but the divide byzero situation when y=0 is avoided. At this point, only real (notcomplex) solutions of x and y are valid, which possibly eliminates twoof the four total solution sets. Control fields can be computed usingequation (18) for each of the remaining valid paired solutions of x andy:

$\begin{matrix}{B_{c} = {\begin{bmatrix}B_{c1} \\B_{c2} \\B_{c3}\end{bmatrix} = {\begin{bmatrix}\frac{2}{3} & 0 \\{- \frac{1}{3}} & \frac{\sqrt{3}}{3} \\{- \frac{1}{3}} & {- \frac{\sqrt{3}}{3}}\end{bmatrix} \cdot \begin{bmatrix}x \\y\end{bmatrix}}}} & (18)\end{matrix}$

Invalid solutions can be eliminated for which any of the total fieldmagnitudes corresponding to each solution set exceed B_(max), which isthe maximum allowed air gap field. Note that this is equivalent tocomparing the L-infinity norm of the total fields vector to B_(max),where the total fields vector is the sum of the control field vector andbias field: |B_(c)+B₀[1,1,1]^(T)|_(∞)≤B_(max). For illustration, anoptimum solution may be selected that minimizes a conduction loss basedon equation (19)

$\begin{matrix}{B_{c,{optimal}} = {\underset{B_{c,{valid}}}{\arg \min}( {B_{c}}_{2} )}} & (19)\end{matrix}$

where B_(c,valid) represents a set of all valid solutions.

The control currents can be computed from the optimal control fieldsusing equation (20).

$\begin{matrix}{\begin{bmatrix}i_{c1} \\i_{c2} \\i_{c3}\end{bmatrix} = {\frac{1}{k_{2}}B_{c,{optimal}}}} & (20)\end{matrix}$

As another option, the control currents can, of course, be computeddirectly as shown below:

$\begin{matrix}{i_{c} = {\begin{bmatrix}i_{c1} \\i_{c2} \\i_{c3}\end{bmatrix} = {{\frac{1}{k_{2}}\begin{bmatrix}\frac{2}{3} & 0 \\{- \frac{1}{3}} & \frac{\sqrt{3}}{3} \\{- \frac{1}{3}} & {- \frac{\sqrt{3}}{3}}\end{bmatrix}} \cdot \begin{bmatrix}x \\y\end{bmatrix}}}} & (21)\end{matrix}$

An optimal solution can be selected based on eliminating any solutionwith a current larger than a maximum allowable coil current, calculatingcorresponding air gap fields and eliminating any solution that wouldexceed B_(max), and, of the remaining solutions, selecting the x, ypaired solution that minimizes an L2 norm of the currents

$i_{c,{optimal}} = {{\underset{i_{c,{valid}}}{\arg \min}( {i_{c}}_{2} )}.}$

In a third option, control current calculations can be made based on anexperimentally measured or pre-calculated force-current relationship.The possibility of doing this can be seen from re-writing equation (1)as

${\overset{arrow}{F} = { {k_{1}k_{2}^{2}}arrow{\overset{->}{\iota_{c}^{2}} + {2B_{0}k_{1}k_{2}\overset{->}{\iota_{c}}}}  = {{C_{1}\overset{->}{\iota_{c}^{2}}} + {C_{2}\overset{->}{\iota_{c}}}}}},$

where C₁=k₁k₂ ² and C₂=2B₀k₁k₂. An example procedure to fit C₁ and C₂consists of generating a force in the positive x-direction withsymmetric three-phase currents of

${{i_{1} = \overset{\hat{}}{I}},{i_{2} = {i_{3} = {- \frac{\hat{I}}{2}}}}}.$

In this case

${\overset{->}{\iota_{c}} = {\frac{3}{2}\hat{I}}},{ arrow\overset{->}{\iota_{c}^{2}}  = {\frac{3}{4}{\overset{\hat{}}{I}}^{2}}},$

resulting in a purely x-directional force solution ofF_(x)=3/4C₁Î²+3/2C₂Î. This force can be measured with a load cell whileÎ is varied and C₁ and C₂ can be fit to the resulting data. Once C₁ andC₂ are obtained, the depressed quartic of equation (8) can be solved inthe same manner as previously described, where effective values of k₁,k₂ and B₀ are determined by solving C₁=k₁k₂ ² and C₂=2B₀k₁k₂. To fullydefine these values, B₀ can either be measured, or an effective value ofone of the variables can simply be assumed. In the case that a value isassumed for any of these variables, k₁, k₂, and B₀ should no longer beregarded as having physical meaning. For example, B₀ would no longer bethe bias field, but rather would simply be a mathematical constant.

Referring to FIG. 30, example operations associated with controlapplication 120 in computing the control currents from the desired forcevector F_(x) and F_(y) for RMB 106 or for the RMB portion of CRAMB 104are described. Additional, fewer, or different operations may beperformed depending on the embodiment of control application 120. Forexample, control application 120 may further compute currents/voltagesor other signals to control operation of motor 102, currents/voltages tocontrol operation of the axial portion of CRAMB 104, etc. The order ofpresentation of the operations of FIG. 30 is not intended to belimiting. Some of the operations may not be performed in someembodiments. Although some of the operational flows are presented insequence, the various operations may be performed in variousrepetitions, concurrently (in parallel, for example, using threads),and/or in other orders than those that are illustrated. Controlapplication 120 may be executed automatically when motor 102 is startedor when power is provided to electrical machine system 100.

In an operation 3000, the area correction factor β is computed usingequation (5) above and radial tooth angle 1000 θ₁ in radians provided asan input value. Alternatively, the area correction factor β may beprovided as an input value. For example, radial tooth angle 1000 θ₁ inradians and/or the area correction factor may be received by controlapplication 120 after selection from a user interface window, afterentry by a user into a user interface window, by extracting theinformation from a request, by reading a value stored incomputer-readable medium 116, etc.

In an operation 3002, the first proportionality constant k₁ and thesecond proportionality constant k₂ are computed using equations (4) and(6), respectively and the computed area correction factor β, the air gaparea A, the permeability of free space μ₀, the number of turns of firstradial coil 808, second radial coil 810, and third radial coil 812, andthe effective radial air gap length 900 g are provided as input values.Alternatively, the first proportionality constant k₁ and the secondproportionality constant k₂ may be provided as an input value. Forexample, the first proportionality constant k₁ and the secondproportionality constant k₂ and/or the air gap area A, the permeabilityof free space μ₀, the number of turns of first radial coil 808, secondradial coil 810, and third radial coil 812, and the effective radial airgap length 900 g may be received by control application 120 afterselection from a user interface window, after entry by a user into auser interface window, by extracting the information from a request, byreading a value stored in computer-readable medium 116, etc. The air gaparea A further may be computed using A=θ₁T, where stator inner radius1004 R and/or stator lamination stack thickness 814 T are provided asinput values.

In an operation 3004, values for the desired force vector F_(x) andF_(y) are received. For example, the values for F_(x) and F_(y) may bereceived and/or computed from sensor measurement from one or moresensors configured to measure radial movement of rotor shaft 1600.

In an operation 3006, the quartic coefficients for the y-component canbe computed using equations (9), (10), and (11) above and the bias fieldB₀, the values for F_(x) and F_(y), and the computed proportionalityconstant k₁. The bias field B₀ may be provided as an input value. Thebias field B₀ equals an average of the three radial pole fields providedby first radial stator lamination tooth 802, second radial statorlamination tooth 804, and third radial stator lamination tooth 806.Inclusion of compensation coil 302 and elimination of zero sequencecurrents, the common mode flux is only dependent on the plurality ofpermanent magnets 204. As a result, the bias field B₀ may be regarded asa permanent magnet bias field. To determine what bias field B₀ is forRMB 106 or of the RMB portion of CRAMB 104, the field can be measured infront of each of first radial stator lamination tooth 802, second radialstator lamination tooth 804, and third radial stator lamination tooth806 and averaged.

Traditionally, for a three-pole bearing, B₀=0 or B₀=1/2B_(max), whereB_(max) is a maximum magnetic field. The maximum magnetic field B_(max)for a bearing is a maximum field in the radial air gap in front of anyof the magnetic poles under any condition. Physically, this can bedetermined based on either a maximum current carrying capability of RMB106 or of the RMB portion of CRAMB 104, a current rating of electricalmachine controller 110, or a magnetic material saturation point,whichever is lower. A simple test that can be used to determine B_(max)is to impart a force on rotor 108 that directly opposes one of firstradial stator lamination tooth 802, second radial stator laminationtooth 804, or third radial stator lamination tooth 806 while measuring afield that occurs directly in front of first radial stator laminationtooth 802, second radial stator lamination tooth 804, or third radialstator lamination tooth 806 in the radial air gap. The force magnitudeis continually increased until RMB 106 or of the RMB portion of CRAMB104 is not able to overcome the imparted force. The maximum field thatis measured over the course of the measurements corresponds to B_(max).

The traditional values for B₀, B₀=0 or B₀=1/2B_(max), are not optimalbias field value for several reasons. Two optimal bias field values havebeen determined as:

$B_{0,{{opt}\; 1}} = {{\frac{1 + \frac{1}{\sqrt{2}}}{3}B_{\max}} \approx {{0.5}690B_{\max}}}$$B_{0,{{opt}\; 2}} = {{\frac{1 - \frac{1}{\sqrt{2}}}{3}B_{\max}} \approx {{0.0}976B_{\max}}}$

A maximum possible rated force can be considered as a radius of a circlethat touches a hexagonal profile at a force angle of α=30° as shownreferring to FIG. 32. At this point, the three pole force componentsmust be as follows

F ₁ =σB ₁ ² =F _(max)

F ₂ =σB ₂ ²=0.5F _(max)

F ₃ =σB ₃ ²=0

where F_(max)=σB_(max) ². Solving for the three radial fields yields:

${B_{1} = B_{\max}}{B_{2} = {{\pm \frac{1}{\sqrt{2}}}B_{\max}}}{B_{3} = 0}$

Again, the bias field B₀ equals the average of the three radial fields.Applying this results in two bias field values:

${\frac{B_{0}}{B_{\max}} = {\zeta = {\frac{1 \pm \frac{1}{\sqrt{2}}}{3} \approx}}}\{ {0.0976,0.5960} \}$

where 0.5960 results with the positive sign and 0.0976 results with thenegative sign. The value using the negative sign requires moreampere-turns compared to the value using the positive sign because eachfield needs to be generated by current in contrast to a permanentmagnet.

The optimal bias results can be validated numerically. Referring to FIG.31, a normalized rated force curve 3100 as a function of the normalizedbias field computed for the RMB portion of CRAMB 104 is shown as thebias field is swept from 0 to B_(max). As expected, the two computedoptimal bias field values are associated with the maximum values ofnormalized rated force curve 3100. If the bias had been selected asζ=0.5, as is typically done in homopolar-biased three pole bearings, thebearing rated bearing force would be reduced by 15.5%. As can be seen inFIG. 31, there are two ranges of bias field values that can be used toincrease a force rating compared to traditional methods:

0<ζ<0.167

0.5<ζ<0.668.

Referring again to FIG. 30, in an operation 3008, the quartic equationin equation (8) is solved to define up to four solutions for they-component denoted y_(k), k=1, . . . , 4.

In an operation 3010, a determination is made concerning whether y_(k)=0for any of k=1, . . . , 4. When y_(k)=0, processing continues in anoperation 3014. When y_(k)≠0, processing continues in an operation 3012.

In operation 3012, up to four solutions for the x-component denotedx_(k), k=1, . . . , 4 are computed using equation (12), and processingcontinues in an operation 3020.

In operation 3014, the quartic coefficients for the x-component can becomputed using equations (14), (15), and (16) above, the bias field B₀,the values for F_(x) and F_(y), and the computed proportionalityconstant

In an operation 3016, the quartic equation in equation (13) is solved todefine up to four solutions for the x-component x_(k), k=1, . . . , 4.

In an operation 3018, up to four solutions for the y-component y_(k),k=1, . . . , 4 are computed using equation (17), and processingcontinues in operation 3020.

In operation 3020, values for the control field for each of first radialcoil 808, second radial coil 810, and third radial coil 812 areoptionally computed using equation (18) for each paired solution of(x_(k),y_(k)), k=1, . . . , 4. The three values for the control field ofeach paired solution of (x_(k),y_(k)), k=1, . . . , 4 define a controlfield set, where up to four control field sets may be computed. Zero ormore of the control field sets may be eliminated as discussed above.

In an operation 3022, values for the control current to each of firstradial coil 808, second radial coil 810, and third radial coil 812 arecomputed for each paired solution (x_(k),y_(k)), k=1, . . . , 4 usingequation (20) or equation (21). The three values for the control currentof each paired solution of (x_(k),y_(k)), k=1, . . . , 4 define acontrol current set, where up to four control current sets may becomputed.

In an operation 3024, values for the control current may be selectedfrom the up to four control current sets as discussed above.

In an operation 3026, the selected control currents are output to eachof first radial coil 808, second radial coil 810, and third radial coil812 to control operation of RMB 106 or the RMB portion of CRAMB 104.Processing may continue in operation 3004 to await updated values forthe desired force vector F_(x) and F_(y).

Using the optimal bias fields results in a 15.5% higher rated forcecompared to using the traditional values of B₀=0 or B₀=B_(max). Thismeans that for an equivalently rated three-pole RMB, rotor 108 is eithershorter in length or smaller in diameter.

Use of the control method described in FIG. 30 or =0.569 or =0.0976,alone or in combination can be applied to improve any three-pole RMBthat uses a common mode bias field. This includes combined radial-axialmagnetic bearings as well as purely radial bearings. The control methoddescribed in FIG. 30 computes an exact inverse force to currentrelationship resulting in significantly reduced force vector error. Theoptimal bias field increases the force density of the bearing resultingin a reduced shaft length to produce a same rated force as other threepole radial magnetic bearings. Further, the three pole radial magneticbearing can be operated using a three phase inverter offeringsignificant cost reduction due to the high production quantities ofmotor inverter products. Each half-bridge of the three phase inverter isconnected to one of the first winding, the second winding, or the thirdwinding wrapped around first radial stator lamination tooth 802, secondradial stator lamination tooth 804, and third radial stator laminationtooth 806, respectively, to provide the selected control current.

As used herein, the term “mount” includes join, unite, connect, couple,associate, insert, hang, hold, affix, attach, fasten, bind, paste,secure, hinge, bolt, screw, rivet, solder, weld, glue, form over, formin, layer, mold, rest on, rest against, abut, and other like terms. Thephrases “mounted on”, “mounted to”, and equivalent phrases indicate anyinterior or exterior portion of the element referenced. These phrasesalso encompass direct mounting (in which the referenced elements are indirect contact) and indirect mounting (in which the referenced elementsare not in direct contact, but are connected through an intermediateelement) unless specified otherwise. Elements referenced as mounted toeach other herein may further be integrally formed together, forexample, using a molding or thermoforming process as understood by aperson of skill in the art. As a result, elements described herein asbeing mounted to each other need not be discrete structural elementsunless specified otherwise. The elements may be mounted permanently,removably, or releasably unless specified otherwise.

Use of directional terms, such as top, bottom, right, left, front, back,upper, lower, horizontal, vertical, behind, etc. are merely intended tofacilitate reference to the various surfaces of the described structuresrelative to the orientations introduced in the drawings and are notintended to be limiting in any manner unless otherwise indicated.

As used in this disclosure, the term “connect” includes join, unite,mount, couple, associate, insert, hang, hold, affix, attach, fasten,bind, paste, secure, bolt, screw, rivet, pin, nail, clasp, clamp,cement, fuse, solder, weld, glue, form over, slide together, layer, andother like terms. The phrases “connected on” and “connected to” includeany interior or exterior portion of the element referenced. Elementsreferenced as connected to each other herein may further be integrallyformed together. As a result, elements described herein as beingconnected to each other need not be discrete structural elements. Theelements may be connected permanently, removably, or releasably.

The word “illustrative” is used herein to mean serving as an example,instance, or illustration. Any aspect or design described herein as“illustrative” is not necessarily to be construed as preferred oradvantageous over other aspects or designs. Further, for the purposes ofthis disclosure and unless otherwise specified, “a” or “an” means “oneor more”. Still further, using “and” or “or” in the detailed descriptionis intended to include “and/or” unless specifically indicated otherwise.

The foregoing description of illustrative embodiments of the disclosedsubject matter has been presented for purposes of illustration and ofdescription. It is not intended to be exhaustive or to limit thedisclosed subject matter to the precise form disclosed, andmodifications and variations are possible in light of the aboveteachings or may be acquired from practice of the disclosed subjectmatter. The embodiments were chosen and described in order to explainthe principles of the disclosed subject matter and as practicalapplications of the disclosed subject matter to enable one skilled inthe art to utilize the disclosed subject matter in various embodimentsand with various modifications as suited to the particular usecontemplated. It is intended that the scope of the disclosed subjectmatter be defined by the claims appended hereto and their equivalents.

What is claimed is:
 1. A non-transitory computer-readable medium havingstored thereon computer-readable instructions that when executed by aprocessor cause a controller to: receive an indicator of a first radialforce value in a first radial force direction and a second radial forcevalue in a second radial force direction, wherein the first radial forcedirection is perpendicular to the second radial force direction; computecoefficients for a first equation using the first radial force value andthe second radial force value, wherein the first equation is anon-linear equation; solve the first equation to define a plurality offirst solution values; compute a second solution value paired with eachfirst solution value of the plurality of first solution values using thefirst radial force value and a respective first solution value to definea plurality of second solution values; compute a plurality of controlcurrent sets, wherein each control current set of the plurality ofcontrol current sets includes a control current value for each of threecontrol currents, wherein the plurality of control current sets includesa control current set computed for each unique paired solution of theplurality of second solution values and the plurality of first solutionvalues; select a control current value for each of the three controlcurrents from a control current set of the computed plurality of controlcurrent sets; and output the selected control current value for each ofthe three control currents to a respective radial winding of athree-pole radial magnetic bearing.
 2. The non-transitorycomputer-readable medium of claim 1, wherein the first equation is aquartic equation defined as y⁴+py²+qy+r=0, where y is an imaginary partof a control field space vector, p is a first coefficient of thecomputed quartic coefficients, q is a second coefficient of the computedquartic coefficients, and r is a third coefficient of the computedquartic coefficients.
 3. The non-transitory computer-readable medium ofclaim 2, wherein p is computed using${p = {3( {\frac{F_{x}}{k_{1}} - {9B_{0}^{2}}} )}},$ whereF_(x) is the second radial force value, k₁ is a predefinedproportionality constant, and B₀ is a predefined bias field value. 4.The non-transitory computer-readable medium of claim 3, wherein B₀ isdefined as 0<B₀<0.167B_(max), where B_(max) is a predefined maximumtotal air gap field value defined for the three-pole radial magneticbearing.
 5. The non-transitory computer-readable medium of claim 4,wherein B₀=0.0976B_(max).
 6. The non-transitory computer-readable mediumof claim 4, wherein B_(max) is defined based on a maximum currentcarrying capability of the three-pole radial magnetic bearing.
 7. Thenon-transitory computer-readable medium of claim 4, wherein B_(max) isdefined based on a magnetic material saturation point of the three-poleradial magnetic bearing.
 8. The non-transitory computer-readable mediumof claim 3, wherein B₀ is defined as 0.5B_(max)<B₀<0.66B_(max), whereB_(max) is a predefined maximum total air gap field value defined forthe three-pole radial magnetic bearing.
 9. The non-transitorycomputer-readable medium of claim 8, wherein B₀=0.5690B_(max).
 10. Thenon-transitory computer-readable medium of claim 8, wherein B_(max) isdefined based on a maximum current carrying capability of the three-poleradial magnetic bearing.
 11. The non-transitory computer-readable mediumof claim 8, wherein B_(max) is defined based on a magnetic materialsaturation point of the three-pole radial magnetic bearing.
 12. Thenon-transitory computer-readable medium of claim 3, wherein k₁ isdefined as${k_{1} = \frac{2{\sin ( \frac{\theta_{1}}{2} )}A}{\theta_{1}\mu_{0}}},$where θ₁ is an angle that each tooth of a stator lamination stack of thethree-pole radial magnetic bearing spans, A is an air gap area of eachtooth, and μ₀ is a permeability of free space.
 13. The non-transitorycomputer-readable medium of claim 2, wherein q is computed using${q = {18B_{0}\frac{F_{y}}{k_{1}}}},$ where F_(y) is the first radialforce value, k₁ is a predefined proportionality constant, and B₀ is apredefined bias field value.
 14. The non-transitory computer-readablemedium of claim 2, wherein r is computed using${r = {{- \frac{9}{4}}( \frac{F_{y}}{k_{1}} )^{2}}},$ whereF_(y) is the first radial force value, and k₁ is a predefinedproportionality constant.
 15. The non-transitory computer-readablemedium of claim 2, wherein the plurality of second solution values arecomputed using${x = {\frac{3}{2}( {{2B_{0}} - {\frac{F_{y}}{k_{1}}( \frac{1}{y} )}} )}},$where F_(y) is the first radial force value, k₁ is a predefinedproportionality constant, and B₀ is a predefined bias field value, and yis one of the plurality of first solution values.
 16. The non-transitorycomputer-readable medium of claim 1, wherein the control current setselected from the computed plurality of control current sets has aminimum L2 norm for the three control currents of the computed pluralityof control current sets.
 17. The non-transitory computer-readable mediumof claim 1, wherein the plurality of control current sets are computedusing ${\begin{bmatrix}i_{1} \\i_{2} \\i_{3}\end{bmatrix} = {{\frac{1}{k_{2}}\begin{bmatrix}\frac{2}{3} & 0 \\{- \frac{1}{3}} & \frac{\sqrt{3}}{3} \\{- \frac{1}{3}} & {- \frac{\sqrt{3}}{3}}\end{bmatrix}}\begin{bmatrix}x_{k} \\y_{k}\end{bmatrix}}},$ where i₁ is a first control current value of thecontrol current set, i₂ is a second control current value of the controlcurrent set, i₃ is a third control current value of the control currentset, k₂ is a predefined proportionality constant, and x_(k),y_(k) are ak^(th) unique paired solution of the plurality of second solution valuesand the plurality of first solution values such that x_(k) is one of theplurality of second solution values, and y_(k) is one of the pluralityof first solution values.
 18. The non-transitory computer-readablemedium of claim 1, wherein the first equation is a quartic equationdefined as x⁴+px²+q+r=0, where x is a real part of a control field spacevector, p is a first coefficient of the computed quartic coefficients, qis a second coefficient of the computed quartic coefficients, and r is athird coefficient of the computed quartic coefficients.
 19. Thenon-transitory computer-readable medium of claim 18, wherein p iscomputed using${p = {{- 3}( {\frac{F_{x}}{k_{1}} + {9B_{0}^{2}}} )}},$ qis computed using${q = {18{B_{0}( {{3B_{0}^{2}} + \frac{F_{x}}{k_{1}}} )}}},$and r is computed using${r = {- ( {{27{B_{0}^{2}( \frac{F_{x}}{k_{1}} )}} + {\frac{9}{4}( \frac{F_{y}}{k_{1}} )^{2}}} )}},$where F_(x) is the first radial force value, F_(y) is the second radialforce value, k₁ is a predefined proportionality constant, and B₀ is apredefined bias field value.
 20. The non-transitory computer-readablemedium of claim 19, wherein the plurality of second solution values arecomputed using${y = {\frac{F_{y}}{k_{1}}( {{2B_{0}} - {\frac{2}{3}x}} )^{- 1}}},$where x is one of the plurality of first solution values.